Question 1058529: A student earns $10 per hour for tutoring and $7 per hour as a teacher's aide. To have enough free time for studies, he can work no more than 20 hours per week. The tutoring center requires that each tutor spends at least three hours per week tutoring, but no more than eight hours per week. How many hours should he work to maximize his earnings?
hours of tutoring
hours as a teacher's aide
What is the maximum profit? $
Answer by solve_for_x(190)   (Show Source):
You can put this solution on YOUR website! There are a few ways to approach this problem.
One method is to treat it as a linear programming problem.
Let x represent the number of hours worked in the tutoring center, and
let y represent the number of hours worked as a teacher's aide.
The profit function is:
P = 10x + 7y
and is subject to the following constraints:
x + y ≤ 20 (because the student cannot work more than 20 hours)
x ≥ 3 (because the tutoring center requires a minimum of 3 hours)
x ≤ 8 (because the tutoring center allows a maximum of 8 hours)
The constraints can be graphed to give the following region:
The corner points of the shaded region are:
(3, 0), (3, 17), (8, 0), and (8, 12)
Evaluating the profit function at each point gives:
P(3, 0) = 10(3) + 7(0) = 30
P(3, 17) = 10(3) + 7(17) = 149
P(8, 0) = 10(8) + 7(0) = 80
P(8, 12) = 10(8) + 7(12) = 164 <--- Maximum
The maximum profit is $164 when the student works the maximum 8 hours allowed
in the tutoring center, and then works 12 hours as a teacher's aide.
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